Real Paley-Wiener Theorem for the Generalized Weinstein transform in quantum calculus (original) (raw)

Paley-Wiener theorems for the Schrodinger operator on BbbR\Bbb RBbbR

1998

summary:In this paper we define and study generalized Fourier transforms associated with some class of Schrodinger operators on BbbR\Bbb RBbbR. Next, we establish Paley-Wiener type theorems which characterize some functional spaces by their generalized Fourier transforms

Real Paley-Wiener theorems for the Koornwinder-Swarttouw q-Hankel transform

Journal of Mathematical Analysis and Applications, 2007

We derive two real Paley–Wiener theorems in the setting of quantum calculus. The first uses techniques due to Tuan and Zayed [VK Tuan, AI Zayed, Paley–Wiener-type theorems for a class of integral transforms, J. Math. Anal. Appl. 266 (1) (2002) 200–226] in order to describe ...

Paley-Wiener theorems for the Schrodinger operator on R

1998

In this paper we define and study generalized Fourier transforms associated with some class of Schrodinger operators on R. Next, we establish Paley-Wiener type theorems which characterize some functional spaces by their generalized Fourier transforms.

Paley–Wiener theorem for the Weinstein transform and applications

Integral Transforms and Special Functions, 2017

In this paper our aim is to establish the Paley-Wiener Theorems for the Weinstein Transform. Furthermore, some applications are presents, in particular some properties for the generalized translation operator associated with the Weinstein operator are proved.

Paley–Wiener subspace of vectors in a Hilbert space with applications to integral transforms

Journal of Mathematical Analysis and Applications, 2009

The goal of this article is to introduce an analogue of the Paley-Wiener space of bandlimited functions, PW ω , in Hilbert spaces and then apply the general result to more specific examples. Guided by the role that the differentiation operator plays in some of the characterizations of the Paley-Wiener space, we construct a space of vectors using a selfadjoint operator D in a Hilbert space H, and denote this space by PW ω (D). The article can be virtually divided into two parts. In the first part we show that the space PW ω (D) has similar properties to those of the space PW ω , including an analogue of the Bernstein inequality and the Riesz interpolation formula. We also develop a new characterization of the abstract Paley-Wiener space in terms of solutions of Cauchy problems associated with abstract Schrödinger equations. Finally, we prove two sampling theorems for vectors in PW ω (D), one of which uses the notion of Hilbert frames and the other is based on the notion of variational splines in H. In the second part of the paper we apply our abstract results to integral transforms associated with singular Sturm-Liouville problems. In particular we obtain two new sampling formulas related to one-dimensional Schrödinger operators with bounded potential.

Paley-Wiener type theorems by transmutations

The Journal of Fourier Analysis and Applications, 2001

The classical Paley-Wiener theorem for functions in L2dx relates the growth of the Fourier transform over the complex plane to the support of the function. In this work we obtain Paley-Wiener type theorems where the Fourier transform is replaced by transforms associated with self-adjoint operators on L 2lz , with simple spectrum, where d lz is a Lebesgue-Stieltjes measure. This is achieved via the use of support preserving transmutations. This result shows that any entire function of finite type and order one is the Fourier transform of an L2x function with compact support. For each A 6 R, the kernel of the Fourier transform, exp(ix)0, is an eigenfunctional of the self-adjoint differential operator 79 :=-i a__ acting in L2x: dx 79 exp(ix)~) = ~. exp(ix)0, x ~ R. Here exp(ix~.) are called eigenfunctionals rather than eigenfunctions, since they do not belong to L2x, and therefore are treated as functionals, see [15]. In this work, we show that the Fourier transform appearing in the Paley-Wiener theorem can be replaced by a transform associated with a self-adjoint operator acting on some Hilbert space L2tz, L2u:={f measurable :llfll2=flf(x)12dlz<oo} where d/z is a Lebesgue-Stieltjes measure generated by a nondecreasing function/z, and supp dp~ = R.

A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces

Mathematische Nachrichten

Here we give a new approach to the Paley-Wiener theorem in a Mellin analysis setting which avoids the use of the Riemann surface of the logarithm and analytical branches and is based on new concepts of polar-analytic function in the Mellin setting and Mellin-Bernstein spaces. A notion of Hardy spaces in the Mellin setting is also given along with applications to exponential sampling formulas of optical physics.

On a general q-Fourier transformation with nonsymmetric kernels

Journal of Computational and Applied Mathematics, 1996

Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a q-Fourier transform by using the continuous q-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey-Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of q-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey-Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.

Paley–Wiener-Type Theorems for a Class of Integral Transforms

Journal of Mathematical Analysis and Applications, 2002

A characterization of weighted L2(I) spaces in terms of their images under various integral transformations is derived, where I is an interval (finite or infinite). This characterization is then used to derive Paley-Wiener-type theorems for these spaces. Unlike the classical Paley-Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the complex plane. The class of integral transformations considered is related to singular Sturm-Liouville boundary-value problems on a half line and on the whole line. 1 1991 Mathematics Subject Classifications. Primary 44A15, 34B24; Secondary 42B10, 33C45